Simulated Marginal Structural Model (MSM) with Inverse Probability Weighting and Lagged Proteins for Longitudinal Protein Analysis of Incident T2D

1 Marginal Structural Models (MSMs)

MSMs estimate the causal effect of time-varying exposures (e.g., protein levels) on outcomes (e.g., T2D). Standard regression models can introduce bias due to:

• Time-varying confounding – Past protein levels influence future levels and T2D risk. MSMs can use Inverse Probability Weighting (IPW) to adjust for these confounders.

• Collider bias – Adjusting for a variable influenced by both the exposure (future protein level) and outcome (e.g., adjusting on baseline protein levels, which may be affected by the same genetics as the future protein level and by preclinical disease, aka, reverse causation) can create spurious associations. MSMs do not inherently correct for collider bias; instead, might avoid it by using lagged variables rather than conditioning on potential colliders?

2 What is a Lagged Variable?

A lagged variable is a past value of the same variable, used as a predictor for its future value.

• For example, instead of adjusting for baseline CRP, we might use CRP from the previous time point (t-1) to account for prior inflammation without introducing collider bias.

• This helps estimate how changes in protein levels over time influence T2D risk while avoiding spurious associations.

MSMs correct for confounding with IPW, while collider bias must be prevented using study design—such as using lagged exposures instead of colliders.

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3 MSM and Lagged Proteins: Handling Confounding & Collider Bias

We use MSM and lagged proteins together to handle both confounding and potential collider bias:

Challenge	Solution	Why It Works
Time-varying confounding	MSM + IPW	Balances confounders over time
Potential collider bias (baseline CRP)	Lagged proteins (CRP at t-1)	Avoids spurious associations

4 How These Methods Work Together

✅ MSMs handle time-varying confounders using IPW.

✅ Lagged proteins replace problematic baseline adjustments to avoid collider bias.

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5 Causal Structures

• Confounding: A confounder influences both the exposure and the outcome, creating a spurious association if not adjusted for (e.g., baseline protein levels affecting both future protein levels and T2D). MSMs can use Inverse Probability Weighting (IPW) to handle confounding.

• Collider: A collider is influenced by both the exposure and the outcome, and adjusting for it can induce a false association (e.g., baseline protein levels affected by genetics and preclinical T2D). Collider bias is avoided by not conditioning on the collider and using alternative strategies like lagged variables.

DAG Code

grViz("
digraph G {
  graph [layout = dot, rankdir = TB]

  # Confounder Scenario
  subgraph cluster_1 {
    label = 'Confounder Scenario'
    CRP0 [label='Baseline CRP', shape=ellipse, style=filled, fillcolor=lightpink]
    T2D [label='Incident T2D', shape=box, style=filled, fillcolor=lightblue]
    CRP1 [label='Future CRP', shape=ellipse, style=filled, fillcolor=lightpink]
    Genetics [label='Genetics/Lifestyle Factors', shape=ellipse, style=filled, fillcolor=lightgray]

    Genetics -> CRP0
    Genetics -> T2D
    CRP0 -> T2D
    CRP0 -> CRP1
  }

  # Collider Scenario
  subgraph cluster_2 {
    label = 'Collider Scenario'
    PreT2D [label='Pre-clinical T2D', shape=box, style=filled, fillcolor=lightblue]
    CRP0_c [label='Baseline CRP', shape=ellipse, style=filled, fillcolor=lightpink]
    CRP1_c [label='Future CRP', shape=ellipse, style=filled, fillcolor=lightpink]
    Genetics_c [label='Genetics/Lifestyle Factors', shape=ellipse, style=filled, fillcolor=lightgray]

    Genetics_c -> CRP0_c
    PreT2D -> CRP0_c
    CRP0_c -> CRP1_c
    Genetics_c -> CRP1_c
  }
}
")

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6 Simulation Example

I simulated a longitudinal dataset with 500 individuals across 4 time points to study the relationship between protein levels and T2D using a MSM with IPW and lagged variables.

6.1 Variables in the Dataset

• id: Unique identifier for each individual.
• time: Time point (1 to 4).
• protein: Protein levels measured at each time point.
• bmi: Time-varying Body Mass Index (BMI).
• activity: Physical activity (binary: 0 = low, 1 = high).
• medication: Medication use (binary: 0 = no, 1 = yes).
• t2d: Type 2 Diabetes onset (binary: 0 = no, 1 = yes).
• protein_lag: Lagged protein levels (previous time point).
• weights: Stabilized Inverse Probability Weights (IPWs) used in the MSM.

6.2 Simulation Details

• Protein levels change over time, influenced by BMI, activity, and medication.
• T2D risk is modeled realistically (~10-20% prevalence).
• We aim to avoid collider bias by excluding baseline protein levels using lagged protein values instead.
• Time-varying confounders (BMI, activity, medication) are included, and IPW adjusts for them.

6.3 Why This Approach?

• Marginal Structural Models (MSMs) allow proper estimation of causal effects in the presence of time-varying confounding.
• Lagged protein values prevent bias from conditioning on baseline protein, which maybe act as a collider.
• IPW balances the dataset, mimicking a randomized trial.

This dataset is structured for causal inference and designed to address biases inherent in traditional regression models. By using lagged variables and IPW, we ensure that the estimated effect of protein levels on T2D is not confounded by prior exposures.

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Show Simulation Code

# Load necessary libraries
library(dplyr)
library(geepack)
library(survival)

# Set seed for reproducibility
set.seed(123)

# Define sample size and number of time points (4 longitudinal measurements)
n <- 500  
time_points <- 4  

# Select 18% of individuals to develop T2D
case_ids <- sample(1:n, size = round(0.18 * n), replace = FALSE)

# Simulate baseline covariates
protein_base <- rnorm(n, 5, 1)  # Baseline protein levels
bmi_base <- rnorm(n, 25, 3)  # Baseline BMI
activity_base <- rbinom(n, 1, 0.5)  # Physical activity (0 = low, 1 = high)
medication_base <- rbinom(n, 1, 0.3)  # Medication use (0 = no, 1 = yes)

# Initialize dataset
data <- data.frame(
    id = rep(1:n, each = time_points),
    time = rep(1:time_points, n),
    protein = NA,
    bmi = NA,
    activity = NA,
    medication = NA,
    t2d = 0  # Default to 0, will be updated
)

# Simulate time-varying covariates and T2D progression
for (i in 1:n) {
    has_t2d <- FALSE  # Track if participant develops T2D
    for (t in 1:time_points) {
        idx <- (i - 1) * time_points + t
        
        # Simulate time-varying covariates
        data$bmi[idx] <- bmi_base[i] + rnorm(1, 0.2 * (t - 1), 1)  # BMI slight increase
        data$activity[idx] <- rbinom(1, 1, 0.5)  # Random activity changes
        data$medication[idx] <- rbinom(1, 1, plogis(0.3 * (t - 1)))  # Increasing medication use
        
        # Simulate protein levels over time
        data$protein[idx] <- protein_base[i] + 
            rnorm(1, 0.5 * (t - 1), 0.2) +  # Small protein increase
            0.5 * data$bmi[idx] - 0.3 * data$activity[idx] + 0.6 * data$medication[idx]
        
        # Adjust probability of T2D onset based on selected case IDs
        if (i %in% case_ids && !has_t2d) {
            prob_t2d <- plogis(-4 + 0.1 * data$protein[idx] + 0.1 * data$bmi[idx] - 
                                   0.1 * data$activity[idx] + 0.3 * data$medication[idx])
            
            # Increase likelihood of progression over time
            has_t2d <- rbinom(1, 1, prob_t2d * (t / time_points))  # Increase over time
        }
        
        data$t2d[idx] <- has_t2d  # Once T2D appears, it persists
    }
}

# Ensure 't2d' is a factor with two levels
data$t2d <- factor(data$t2d, levels = c("0", "1"))

# Compute lagged protein levels
data <- data %>%
    group_by(id) %>%
    arrange(time) %>%
    mutate(protein_lag = lag(protein, order_by = time)) %>%
    ungroup() %>%
    filter(!is.na(protein_lag))  # Remove first time point (no lagged value)

# Check updated T2D prevalence
prop.table(table(data$t2d))

# Fit weight model (including time-varying confounders)
weight_model <- glm(t2d ~ protein_lag + bmi + activity + medication, 
                    data = data, family = binomial, control = list(maxit = 200))

# Compute stabilized inverse probability weights (IPWs)
p_t2d <- predict(weight_model, type = "response")
p_t2d <- pmax(pmin(p_t2d, 0.99), 0.01)  # Avoid extreme probabilities

data$weights <- with(data, ifelse(t2d == "1", mean(as.numeric(t2d)) / p_t2d, 
                                  (1 - mean(as.numeric(t2d))) / (1 - p_t2d)))

# Ensure weights are not extreme
data$weights <- pmax(pmin(data$weights, 10), 0.1)  # Trim extreme values

# Convert weights to integer for binomial GLM
data$weights <- as.integer(round(data$weights))

# Convert t2d to numeric for binomial glm
data$t2d_numeric <- as.numeric(data$t2d) - 1

# Ensure id is a factor for grouping in GEE models
data$id <- factor(data$id)

# Fit MSM using Generalized Estimating Equations (GEE)
msm <- geeglm(t2d_numeric ~ protein_lag, 
              data = data, 
              id = id, 
              weights = weights, 
              family = binomial, 
              corstr = "exchangeable", 
              control = geese.control(maxit = 50))  # Increase iterations for convergence

# Summary of MSM results
summary(msm)

# Check distribution of protein lag
summary(data$protein_lag)
hist(data$protein_lag, breaks = 30, main = "Distribution of Protein Lag")

# Check T2D progression over time
table(data$time, data$t2d)

# Check individual-level T2D prevalence
length(unique(data$id[data$t2d == "1"]))

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7 Conclusion

• MSMs with IPW help address time-varying confounding.
• Lagged proteins replace baseline protein levels to avoid collider bias.

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8 Caveat

• As I said in lab meeting, I would consult a statistician with more experience than myself.

• I’ve never implemented the approach I outlined—it’s purely hypothetical, and I could be mistaken. The model didn’t converge, but that issue isn’t specific to MSM. It’s unclear to me whether baseline proteins act as confounders or colliders—-that may depend on the protein itself.

• I’m aware you may have already considered these things 😊

• So, if you find it stimulating or useful, that’s why I’m sharing. I think inverse probability weighting (IPW) is worth considering for analyses involving repeat measures.